What is the slope of the linear equation that has the values shown in the table?
| x | y |
| --- | --- |
| -4 | 15 |
| -2 | 9 |
| 0 | 3 |
| 2 | -3 |
| 4 | -9 |
Answer (2)
The slope of the linear equation derived from the given table is − 3 . This was calculated using two points from the table and verified with another set of points. The consistency of the slope confirms the relationship between the variables x and y.
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Identify two points ( x 1 , y 1 ) and ( x 2 , y 2 ) from the table.
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Calculate the slope using the chosen points: m = − 2 − ( − 4 ) 9 − 15 = − 3 .
The slope of the linear equation is − 3 .
Explanation
Understanding the Problem We are given a table of x and y values that represent a linear equation. Our goal is to find the slope of this line. The slope, often denoted as m , represents the rate of change of y with respect to x . In other words, it tells us how much y changes for every unit change in x .
The Slope Formula To find the slope, we can use the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are any two points on the line.
Calculating the Slope Let's choose the points ( − 4 , 15 ) and ( − 2 , 9 ) from the table. Plugging these values into the slope formula, we get: m = − 2 − ( − 4 ) 9 − 15 = − 2 + 4 − 6 = 2 − 6 = − 3 So, the slope is − 3 .
Verifying the Slope To verify our result, let's choose two different points, say ( 0 , 3 ) and ( 2 , − 3 ) . Using the slope formula again: m = 2 − 0 − 3 − 3 = 2 − 6 = − 3 The slope is still − 3 , which confirms our previous calculation.
Final Answer Therefore, the slope of the linear equation is − 3 .
Examples
Understanding slope is crucial in many real-world applications. For example, if you're analyzing the relationship between the number of hours studied and the score on a test, the slope tells you how much the test score increases for each additional hour of studying. Similarly, in business, if you're looking at the relationship between advertising spending and sales, the slope indicates how much sales increase for each dollar spent on advertising. These concepts are also used in fields like physics (velocity as the slope of a position-time graph) and economics (marginal cost as the slope of a cost-quantity graph).
